South Yorkshire police chief constable David Crompton claimed the corporation appeared to "distance itself from what had happened" after it broadcast the raid on the pop star's penthouse.The BBC has been heavily criticised for its handling of the story.
Mr Crompton said the BBC's misleading and inaccurate portrayal of its involvement in what happened led to the force making a formal complaint.
Keith Vaz, home affairs committee chairman, said the claims were a "matter of deep concern", and summoned the chief constable and BBC director general Tony Hall to appear before the committee next month.
The singer's apartment was searched as part of an investigation into an alleged sexual assault on a young boy at a religious event in 1985. Sir Cliff, who was in Portugal when the search took place, has firmly denied any wrongdoing.